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According to this concept, if wave breaking dominates the dynamics and thus defines the
spectral shape at certain scales,
f
−
5
behaviour of the spectrum should be expected.
In this regard, the transition from
f
−
4
to
f
−
5
can be interpreted as transition from
inherent-breaking (nonlinear fluxes form the spectrum) to induced-breaking (breaking
forms the spectrum) domination. In relative terms, typically such a transition would
occur at
f
t
≈
3
f
p
.
(5.42)
At the end of this subsection, we should mention that expressions/parameterisations of
breaking rates
b
T
(
(2.4)
or
(5.28)
are not the only way to characterise the breaking prob-
ability across the spectrum.
Filipot
et al.
(
2010
), for example, provided a parameterisation
of breaking wave height distributions (BWHD) within selected frequency (wavenumber)
bins.
The target of
Filipot
et al.
(
2010
) was the dissipation function
S
ds
(
f
)
in
(2.61)
. There-
fore, like everybody before them, they had to deal with the choice of the spectral bandwidth
(2.5)
for converting the physical wave count into spectral values. For convenience of
comparisons with the experimental outcomes available for the spectral peak, they chose
f
)
3asin
(2.6)
of
Banner
et al.
(
2000
) and
Babanin
et al.
(
2001
), smoothed by a
Hann window.
This windowed band was then applied across the spectrum. It is quite broad, and, for
example, in the parts of the spectra characterised by the
f
−
4
or
f
−
5
behaviour the wave
height attributed to such bands will be biased towards wave heights at a lower-frequency
end of a respective spectral bin. For
Filipot
et al.
(
2010
), such a bias was undesirable since
they expected the dissipation rates to be a nonlinear function of wave height. This is why
the BWHD, rather than the bin-averaged breaking rates
b
T
(
f
=
0
.
)
f
(2.4)
or
(5.28)
, were chosen
as a characteristic for the breaking probability.
Following
Thornton & Guza
(
1983
), they introduced a formulation for BWHD based on
a Rayleigh distribution
p
(
H
)
(3.43)
of non-breaking wave heights:
p
B
(
H
)
=
p
(
H
)
W
(
H
),
(5.43)
where the weighting function
W
is subject to choice, research and calibration.
Filipot
et al.
(
2010
) targeted a universal deep-to-shallow-water function, which if converted into
deep-water conditions, was chosen as
(
H
)
a
k
r
(
2
1
exp
kH
β
p
f
c
)
H
r
(
f
c
)
W
(
H
,
f
c
)
=
−
−
.
(5.44)
β
β
=
Here,
a
=
1
.
5 and
p
=
4 are tuned parameters,
0
.
25-0
.
3 was found empirically, and
k
r
and
H
r
=
H
rms
are average wavenumber and wave height for the bins centered at the fre-
quency
f
c
within the Hann window, respectively. In finite depths, the function is somewhat
more complicated:
a
β
r
β
2
1
exp
β
β
p
W
(
H
,
f
c
)
=
−
−
.
(5.45)
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